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H. Poincaré, Sur une théorème de M. Liapounoff relatif a l’équilibre d’une masse fluide, Comptes Rendus de L’Academie des Sciences 104, 622–625 (1887).

As discussed by G.C. Evans, Poincaré assumes tacitly that there do exist one or more bodies of a given volume, with smooth boundaries, which provide relative minima for the Coulomb energy with respect to neighboring forms; and his treatment amounts to a proof that among these the sphere provides the absolute minimum.

A complete proof, without this assumption, was given by G. Szegö, Über einige Extremalaufgaben der Potentialtheorie (1930).

H. Poincaré, Sur une théorème de M. Liapounoff relatif a l’équilibre d’une masse fluide, Comptes Rendus de L’Academie des Sciences 104, 622–625 (1887).

As discussed by G.C. Evans, Poincaré assumes tacitly that there do exist one or more bodies of a given volume, with smooth boundaries, which provide relative minima for the electrostatic capacity $C$ with respect to neighboring forms; and his treatment amounts to a proof that among these the sphere provides the absolute minimum. (Note that the Coulomb energy $U=Q^2/2C$, so minimal $C$ corresponds to maximal $U$ for given total charge $Q$.)

A complete proof, without this assumption, was given by G. Szegö, Über einige Extremalaufgaben der Potentialtheorie (1930).

H. Poincaré, Sur une théorème de M. Liapounoff relatif a l’équilibre d’une masse fluide, Comptes Rendus de L’Academie des Sciences 104, 622–625 (1887).

As discussed by G.C. Evans, Poincaré assumes tacitly that there do exist one or more bodies of a given volume, with smooth boundaries, which provide relative minima for the Coulomb energy with respect to neighboring forms; and his treatment amounts to a proof that among these the sphere provides the absolute minimum.

H. Poincaré, Sur une théorème de M. Liapounoff relatif a l’équilibre d’une masse fluide, Comptes Rendus de L’Academie des Sciences 104, 622–625 (1887).

As discussed by G.C. Evans, Poincaré assumes tacitly that there do exist one or more bodies of a given volume, with smooth boundaries, which provide relative minima for the Coulomb energy with respect to neighboring forms; and his treatment amounts to a proof that among these the sphere provides the absolute minimum.

A complete proof, without this assumption, was given by G. Szegö, Über einige Extremalaufgaben der Potentialtheorie (1930).

Poincaré's paper is

H. Poincaré, Sur une theoreme de M. Liapunoff relatif a l’equilibre d’une masse fluide, Comptes Rendus de L’Academie des Sciences 104, 622–625 (1887).

The first complete proof is due to

E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math. 57, 93–105 (1977).

As discussed by G.C. Evans, Poincaré assumes tacitly that there do exist one or more bodies of a given volume, with smooth boundaries, which provide relative minima for the Coulomb energy with respect to neighboring forms; and his treatment amounts to a proof that among these the sphere provides the absolute minimum.

H. Poincaré, Sur une théorème de M. Liapounoff relatif a l’équilibre d’une masse fluide, Comptes Rendus de L’Academie des Sciences 104, 622–625 (1887).

As discussed by G.C. Evans, Poincaré assumes tacitly that there do exist one or more bodies of a given volume, with smooth boundaries, which provide relative minima for the Coulomb energy with respect to neighboring forms; and his treatment amounts to a proof that among these the sphere provides the absolute minimum.